Abstract
Abstract We find global solutions of algebro geometric type for all the equations of a new commuting hierarchy containing the Camassa-Holm equation. This hierarchy is built in analogy to the classical K-dV and AKNS hierarchies. We use a zero curvature method to give recursion formulas. The time evolution of the solutions is completely determined, and the motion on a nonlinear subvariety Υ of a generalized Jacobian variety is obtained by solving an inverse problem for the Sturm-Liouville equation L(φ) = −φ″ + φ = λyφ. This is the natural setting for the expression of the solutions which depend linearly with respect to t and x, with coordinates on a curvilinear parallelogram contained in such a subvariety φ. φ is obtained as the restriction of the generalized Abel map I0 to the space Symmg(R) of unordered g-tuples of points on R, and the nonlinear parallelogram is the image through the restricted generalized Abel map of the moving poles Pi(x, t) (i = 1, . . . , g) of the Weyl m-functions m±(x, t, λ) of the dynamical Sturm-Liouville family of equations Lx(φ) = −φ″ + φ = λτx(y)φ, where τ is the translation flow. It turns out that the choice of a particular stationary initial condition ug(x, t0) completely determines the solution u(x, t) of all the equations in the hierarchy, as functions of the poles Pi(x, t) of the Weyl m-functions corresponding to the family Lx of Sturm-Liouville operators, with density function y(x, t) = uxx(x, t)/2 − 2u(x, t). For every t∊ℝ, the maps x ↦ y(x, t) lie in an isospectral class of the associated family of Sturm-Liouville equations Lx(φ), and are completely determined by assigning spectral parameters and initial conditions for the poles Pi(x, t).
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.